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Chi Square Random Variable

Posted By __Krishna Sankar__ On July 28, 2008 @ 6:14 am In __DSP__ | __12 Comments__

While trying to derive the theoretical bit error rate (BER) for BPSK modulation in a Rayleigh fading channel, I realized that I need to discuss **chi square random variable** prior.

Let there be independent and identically distributed Gaussian random variables with mean and variance and we form a new random variable,

.

Then is a **chi square random variable** with degrees of freedom.

There are two types of **chi square** distribution. The first is obtained when has a zero mean and is called **central chi square distribution**. The second is obtained when has a non-zero mean and is called **non-central chi square distribution**. Four our discussion, we will focus only on central chi square distribution.

Using the text in Chapter 2 of [DIGITAL-COMMUNICATION: PROAKIS] ^{[1]} as reference.

The most simple example of a chi square random variable is

,

where

is a Gaussian random variable with zero mean and variance .

The PDF of is

.

By definition, the cumulative distribution function ^{[2]}(CDF) of is

.

This simplifies to

.

Differentiating the above equation with respect to to find the probability density function,

.

Summarizing, the **pdf of chi square random variable with one degree of freedom** is,

.

Chi square random variable with 2 degrees of freedom is,

,

where,

and are independent Gaussian random variables with zero mean and variance .

In the post on Rayleigh random variable, we have shown that PDF of the random variable,

where is

.

For our current analysis, we know that

.

Differentiating both sides,

.

Applying this to the above equation, **pdf of chi square random variable with two degrees of freedom** is,

.

The probability density function is,

, where

the Gamma function is defined as,

,

p an integer > 0

.

I do not know the proof for deriving the above equation. If any one of you know of good references, kindly let me know. Thanks.

Just for your reference, Matlab/Octave simulation model performing the following is provided

(a) Generate chi square random variables having m=1, 2, 3, 4, 5 degrees of freedom

(b) Probability density function is computed and plotted

Click here to download: Matlab/Octave script for simulating PDF of chi square random variable ^{[3]}

**Figure: PDF of chi square random variable (=1)**

[DIGITAL-COMMUNICATION: PROAKIS] ^{[1]} Digital Communications, by John Proakis ^{[1]}

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URLs in this post:

[1] [DIGITAL-COMMUNICATION: PROAKIS]: **http://www.amazon.com/gp/redirect.html?ie=UTF8&location=http%3A%2F%2Fwww.amazon.com%2FDigital-Communications-John-Proakis%2Fdp%2F0072321113&tag=dl04-20&linkCode=ur2&camp=1789&creative=9325**

[2] cumulative distribution function : **http://en.wikipedia.org/wiki/Cumulative_distribution_function**

[3] Matlab/Octave script for simulating PDF of chi square random variable: **http://www.dsplog.com/db-install/wp-content/uploads/2008/07/pdf_chi_square_random_variable.m**

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